Question

A cop claims you passed him at more than 60 mph ,27.8 meters per second, 200 meters from a red light that just turned green. you drive a car that accelerates at a max of 1.5 meters per second. Is the cop correct?

Answer 1

Sounds like the situation is modeled like so:

[Light] - - - - - - - - - - [You] => - - -

- - - - - - - - - - - - - - - [Cop] - - - - -

<= = = 200 m = = =>

where your velocity is 27.8 m/s to the right. So the question is asking, is it possible that your car, given that it accelerates at a maximum of 1/5 m/s^2, can reach a speed of 27.8 m/s from rest in the time it takes for it to cover 200 m.

The car's position at time $t$ is

$x=\dfrac12\left(1.5\dfrac{\rm m}{\mathrm s^2}\right)t^2$

The time it takes to traverse 200 m is

$200\,\mathrm m=\left(0.75\dfrac{\rm m}{\mathrm s^2}\right)t^2\implies t\approx16.3\,\mathrm s$

Your car's velocity at time $t$, starting from rest, is

$v=\left(1.5\dfrac{\rm m}{\mathrm s^2}\right)t$

so that after 16.3 s, your car is moving at a speed of

$v=\left(1.5\dfrac{\rm m}{\mathrm s^2}\right)\left(16.3\,\mathrm s\right)\approx24.5\dfrac{\rm m}{\rm s}$

which is less than the speed the cop claims you were going, so the cop is not correct.